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NASA NTRS1980

A new approach in cascade flow analysis using the finite element method

A new approach in analyzing the potential flow past cascades and single airfoils using the finite element method is developed. In this analysis the circulation around the airfoil is not externally imposed but is directly computed in the numerical solution. Different finite element discretization patterns, orders of piecewise approximation, and grid sizes are used in the solution. The results obtained are compared with existing experimental measurements and exact solutions in cascades and single airfoils.

Baskharone, E.↗

NASA NTRS1984

Two-dimensional computer simulation of EMVJ and grating solar cells under AMO illumination

A computer program, SCAP2D (Solar Cell Analysis Program in 2-Dimensions), is used to evaluate the Etched Multiple Vertical Junction (EMVJ) and grating solar cells. The aim is to demonstrate how SCAP2D can be used to evaluate cell designs. The cell designs studied are by no means optimal designs. The SCAP2D program solves the three coupled, nonlinear partial differential equations, Poisson's Equation and the hole and electron continuity equations, simultaneously in two-dimensions using finite differences to discretize the equations and Newton's Method to linearize them. The variables solved for are the electrostatic potential and the hole and electron concentrations. Each linear system of equations is solved directly by Gaussian Elimination. Convergence of the Newton Iteration is assumed when the largest correction to the electrostatic potential or hole or electron quasi-potential is less than some predetermined error. A typical problem involves 2000 nodes with a Jacobi matrix of order 6000 and a bandwidth of 243.

Gray, J. L.↗

NASA NTRS1985

A mixed pseudospectral/finite difference method for a thermally driven fluid in a nonuniform gravitational field

A numerical study of the steady, axisymmetric flow in a heated, rotating spherical shell is conducted to model the Atmospheric General Circulation Experiment (AGCE) proposed to run aboard a later shuttle mission. The AGCE will consist of concentric rotating spheres confining a dielectric fluid. By imposing a dielectric field across the fluid a radial body force will be created. The numerical solution technique is based on the incompressible Navier-Stokes equations. In the method a pseudospectral technique is based on the incompressible Navier-Stokes equations. In the method a pseudospectral technique is used in the latitudinal direction, and a second-order accurate finite difference scheme discretizes time and radial derivatives. This paper discusses the development and performance of this numerical scheme for the AGCE which has been modelled in the past only by pure FD formulations. In addition, previous models have not investigated the effect of using a dielectric force to simulate terrestrial gravity. The effect of this dielectric force on the flow field is investigated as well as a parameter study of varying rotation rates and boundary temperatures. Among the effects noted are the production of larger velocities and enhanced reversals of radial temperature gradients for a body force generated by the electric field.

Macaraeg, M. G.↗

NASA NTRS1985

A mixed pseudospectral/finite difference method for a thermally driven fluid in a nonuniform gravitational field

A numerical study of the steady, axisymmetric flow in a heated, rotating spherical shell is conducted to model the Atmospheric General Circulation Experiment (AGCE) proposed to run aboard a later Shuttle mission. The AGCE will consist of concentric rotating spheres confining a dielectric fluid. By imposing a dielectric field across the fluid a radial body force will be created. The numerical solution technique is based on the incompressible Navier-Stokes equations. In the method a pseudospectral technique is used in the latitudinal direction, and a second-order accurate finite difference scheme discretizes time and radial derivatives. This paper discusses the development and performance of this numerical scheme for the AGCE which has been modeled in the past only by pure FD formulations. In addition, previous models have not investigated the effect of using a dielectric force to simulate terrestrial gravity. The effect of this dielectric force on the flow field is investigated as well as a parameter study of varying rotation rates and boundary temperatures. Among the effects noted are the production of larger velocities and enhanced reversals of radial temperature gradients for a body force generated by the electric field.

Macaraeg, M. G.↗

NASA NTRS1986

A computer code for three-dimensional incompressible flows using nonorthogonal body-fitted coordinate systems

In this report, a numerical method for solving the equations of motion of three-dimensional incompressible flows in nonorthogonal body-fitted coordinate (BFC) systems has been developed. The equations of motion are transformed to a generalized curvilinear coordinate system from which the transformed equations are discretized using finite difference approximations in the transformed domain. The hybrid scheme is used to approximate the convection terms in the governing equations. Solutions of the finite difference equations are obtained iteratively by using a pressure-velocity correction algorithm (SIMPLE-C). Numerical examples of two- and three-dimensional, laminar and turbulent flow problems are employed to evaluate the accuracy and efficiency of the present computer code. The user's guide and computer program listing of the present code are also included.

Chen, Y. S.↗

NASA NTRS1986

A numerical method for three-dimensional incompressible flows using nonorthogonal body-fitted coordinate systems

In the present paper, a numerical method for solving the equations of motion of three-dimensional incompressible flows in nonorthogonal body-fitted coordinate (BEC) systems has been developed and evaluated. The equations of motion are transformed to a generalized curvilinear coordinate system from which the transformed equations are discretized using finite difference approximations in the transformed domain. The hybrid scheme and a central differencing plus artificial dissipation scheme are used to approximate the convection terms in the governing equations. Effects of these two schemes on the accuracy of numerical predictions are studied. Solutions of the finite difference equations are obtained iteratively by using a pressure-velocity correction algorithm, SIMPLE-C. Numerical examples of two- and three-dimensional, laminar and turbulent flow problems are employed to evaluate the accuracy and efficiency of the present numerical method.

Chen, Y. S.↗

NASA NTRS1985

A fully implicit scheme for global numerical weather prediction

A fast-slow factored scheme is presented for use with shallow-water primitive equation numerical weather prediction models. The technique was developed to reduce the rotational mode errors which arise when the fast and slow terms of the governing differential equations are treated simultaneously. The method factors out the fast and slow terms along the coordinate directions by means of a modified Crank-Nicolson scheme. A finite-difference spatial discretization is carried out in the zonal and meridional directions to reduce the factorization error to near-zero, and that time steps of 60-90 min can be used to obtain acceptably accurate results, even in the presence of fine spatial structures in the flow.

Augenbaum, J. M.↗

NASA NTRS1988

Numerical analysis of three-dimensional viscous internal flows

A 3-D Navier-Stokes code has been developed for analysis of turbomachinery blade rows and other internal flows. The Navier-Stokes equations are written in a Cartesian coordinate system rotating about the x-axis, and then mapped to a general body-fitted coordinate system. Streamwise viscous terms are neglected using the thin-layer assumption, and turbulence effects are modeled using the Baldwin-Lomax turbulence model. The equations are discretized using finite differences on stacked C-type grids and are solved using a multistage Runge-Kutta algorithm with a spatially-varying time step and implicit residual smoothing. Calculations have been made of a horseshoe vortex formed in front of a flat plate with a round leading edge standing in a turbulent endwall boundary layer. Comparisons are made with experimental data taken by Eckerle and Langston for a circular cylinder under similar conditions. Computer and measured results are compared in terms of endwall flow visualization pictures and total pressure loss contours and vector plots on the symmetry plane. Calculated details of the primary vortex show excellent agreement with the experimental data. The calculations also show a small secondary vortex not seen experimentally.

Chima, Rodrick V.↗

NASA NTRS1988

Numerical analysis of three-dimensional viscous internal flows

Chima, Rodrick V.↗

NASA NTRS1989

Analysis of three-dimensional viscous flow in a supersonic throughflow fan

A 3-D Navier-Stokes code was developed for analysis of turbomachinery blade rows and other internal flows. The Navier-Stokes equations are written in a Cartesian coordinate system rotating about the x-axis, and then mapped to a general body-fitted coordinate system. Streamwise viscous terms are neglected using the thin layer assumption, and turbulence effects are modelled using the Baldwin-Lomax turbulence model. The equations are discretized using finite differences on stacked C-type grids and are solved using a multistage Runge-Kutta algorithm with a spatially varying time step and implicit residual smoothing. Calculations were made of the flow around a supersonic throughflow fan blade. The fan was designed as a key component in a supersonic cruise engine. The 3-D calculations were done on a 129x29x33 grid and took 50 minutes of cpu time. Comparisons with the quasi-3-D results show minor differences in loading due to 3-D effects. Particle traces show nearly 2-D flows near the pressure surface, but large secondary flows within the suction surface boundary layer. The horseshoe vortex ahead of the leading edge is clearly seen.

Chima, Rodrick V.↗

NASA NTRS1993

Application of incremental unknowns to the Burgers equation

In this article, we make a few remarks on the role that attractors and inertial manifolds play in fluid mechanics problems. We then describe the role of incremental unknowns for approximating attractors and inertial manifolds when finite difference multigrid discretizations are used. The relation with direct numerical simulation and large eddy simulation is also mentioned.

Choi, Haecheon↗

NASA NTRS1993

Analysis of multigrid methods on massively parallel computers: Architectural implications

We study the potential performance of multigrid algorithms running on massively parallel computers with the intent of discovering whether presently envisioned machines will provide an efficient platform for such algorithms. We consider the domain parallel version of the standard V cycle algorithm on model problems, discretized using finite difference techniques in two and three dimensions on block structured grids of size 10(exp 6) and 10(exp 9), respectively. Our models of parallel computation were developed to reflect the computing characteristics of the current generation of massively parallel multicomputers. These models are based on an interconnection network of 256 to 16,384 message passing, 'workstation size' processors executing in an SPMD mode. The first model accomplishes interprocessor communications through a multistage permutation network. The communication cost is a logarithmic function which is similar to the costs in a variety of different topologies. The second model allows single stage communication costs only. Both models were designed with information provided by machine developers and utilize implementation derived parameters. With the medium grain parallelism of the current generation and the high fixed cost of an interprocessor communication, our analysis suggests an efficient implementation requires the machine to support the efficient transmission of long messages, (up to 1000 words) or the high initiation cost of a communication must be significantly reduced through an alternative optimization technique. Furthermore, with variable length message capability, our analysis suggests the low diameter multistage networks provide little or no advantage over a simple single stage communications network.

Matheson, Lesley R.↗

NASA NTRS1996

Numerical Study of Multigrid Methods with Various Smoothers for the Elliptical Grid Generation Equations

A robust solver for the elliptic grid generation equations is sought via a numerical study. The system of PDEs is discretized with finite differences, and multigrid methods are applied to the resulting nonlinear algebraic equations. Multigrid iterations are compared with respect to the robustness and efficiency. Different smoothers are tried to improve the convergence of iterations. The methods are applied to four 2D grid generation problems over a wide range of grid distortions. The results of the study help to select smoothing schemes and the overall multigrid procedures for elliptic grid generation.

Golik, W. L.↗

NASA NTRS1999

Physically-Derived Dynamical Cores in Atmospheric General Circulation Models

The algorithm chosen to represent the advection in atmospheric models is often used as the primary attribute to classify the model. Meteorological models are generally classified as spectral or grid point, with the term grid point implying discretization using finite differences. These traditional approaches have a number of shortcomings that render them non-physical. That is, they provide approximate solutions to the conservation equations that do not obey the fundamental laws of physics. The most commonly discussed shortcomings are overshoots and undershoots which manifest themselves most overtly in the constituent continuity equation. For this reason many climate models have special algorithms to model water vapor advection. This talk focuses on the development of an atmospheric general circulation model which uses a consistent physically-based advection algorithm in all aspects of the model formulation. The shallow-water model is generalized to three dimensions and combined with the physics parameterizations of NCAR's Community Climate Model. The scientific motivation for the development is to increase the integrity of the underlying fluid dynamics so that the physics terms can be more effectively isolated, examined, and improved. The expected benefits of the new model are discussed and results from the initial integrations will be presented.

Rood, Richard B.↗

NASA NTRS1999

Physically-Derived Dynamical Cores in Atmospheric General Circulation Models

The algorithm chosen to represent the advection in atmospheric models is often used as the primary attribute to classify the model. Meteorological models are generally classified as spectral or grid point, with the term grid point implying discretization using finite differences. These traditional approaches have a number of shortcomings that render them non-physical. That is, they provide approximate solutions to the conservation equations that do not obey the fundamental laws of physics. The most commonly discussed shortcomings are overshoots and undershoots which manifest themselves most overtly in the constituent continuity equation. For this reason many climate models have special algorithms to model water vapor advection. This talk focuses on the development of an atmospheric general circulation model which uses a consistent physically-based advection algorithm in all aspects of the model formulation. The shallow-water model of Lin and Rood (QJRMS, 1997) is generalized to three dimensions and combined with the physics parameterizations of NCAR's Community Climate Model. The scientific motivation for the development is to increase the integrity of the underlying fluid dynamics so that the physics terms can be more effectively isolated, examined, and improved. The expected benefits of the new model are discussed and results from the initial integrations will be presented.

Rood, Richard B.↗

NASA NTRS1990

Application of implicit numerical techniques to the solution of the three-dimensional diffusion equation

Implicit techniques for calculating three-dimensional, time-dependent heat diffusion in a cube are tested with emphasis on storage efficiency, accuracy, and speed of calculation. For this purpose, a tensor product technique with both Chebyshev collocation and finite differences and a generalized conjugate gradient technique with finite differences are used in conjunction with Crank-Nicolson discretization. An Euler explicit finite difference calculation is performed for use as a benchmark. The implicit techniques are found to be competitive with the Euler explicit method in terms of storage efficiency and speed of calculation and offer advantages both in accuracy and stability. Mesh stretching in the finite difference calculations is shown to markedly improve the accuracy of the solution.

Peltier, Leonard Joel↗

NASA NTRS1986

Polynomial elimination theory and non-linear stability analysis for the Euler equations

Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.

Kennon, S. R.↗

NASA NTRS1984

Dynamic characteristics of a hydrostatic gas bearing driven by oscillating exhaust pressure

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference aproximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

Watkins, C. B.↗